Description and classification of geometric forms have occupied mathematical thinkers since ancient times. The Greeks discovered the five Platonic polyhedra (tetrahedron, cube or hexahedron, octahedron, dodecahedron, and icosahedron) and the thirteen Archimedean polyhedra, all with regular (i.e., equiangular and equilateral) faces. Kepler, who rediscovered the Archimedean polyhedra, discovered two rhombic polyhedra, including the rhombic dodecahedron that resembles ferritin cages. These three classes of polyhedra represent all of the equilateral convex polyhedra with polyhedral symmetry, i.e., icosahedral, octahedral and tetrahedral symmetry. For example, none of the well-known face-regular Johnson solids have polyhedral symmetry.
In a paper titled “A class of multi-symmetric polyhedral,” published in the Tohoku Mathematical Journal 43:104-108 (1937), which is hereby incorporated by reference, the mathematician Michael Goldberg disclosed a novel method for constructing cages with tetrahedral, octahedral, and icosahedral symmetry.
A method for constructing a Goldberg cage is illustrated in FIGS. 1A-1F. First a “Goldberg triangle” is constructed or selected. For example, an equilateral triangle is drawn or positioned on a tiling of hexagons with the vertices of the triangle on the centers of hexagons in the tiling. Examples of suitable Goldberg triangles are shown in FIGS. 1A-1C, wherein the vertices from the tiling that are enclosed by the triangle are shown with a solid circle and vertices that the triangle overlies are shown with a half-filled circle. One edge of the triangle is herein referred to as the “base line segment.”
In general, the base line segment spans h tiles in the horizontal direction (in FIGS. 1A-1C) and k tiles in a direction 60 degrees from horizontal. For example, in the three examples shown in FIG. 1A the base line segment spans h=1, h=2, and h=3 tiles respectively in the horizontal direction, and zero tiles in the 60 degrees direction. In FIG. 1B (left) the base line segment spans h=1 tile in the horizontal direction and k=1 tile in the 60 degree direction. In FIG. 1B (right) the base line segment spans h=2 tiles in the horizontal direction and k=2 tiles in the 60 degree direction. In FIG. 1C (left) the base line segment spans h=2 tiles in the horizontal direction, and k=1 tile in the 60 degree direction. In FIG. 1C (right) the base line segment spans h=3 tiles in the horizontal direction, and k=1 tile in the 60 degree direction.
A Goldberg triangle encloses T vertices (vertices the triangle overlies are counted as ½ an enclosed vertex), where:T=h2+hk+k2  (1)
In FIGS. 1A-1C the figures are labeled with the number of enclosed vertices, T and the (h,k) parameters. Goldberg triangles can be grouped into three different types: (i) the (h,0) group, i.e., k=0 (exemplary embodiments shown in FIG. 1A for T=1, 4 and 9), (ii) the (h=k) group (exemplary embodiments shown in FIG. 1B for T=3 and 12), and (iii) the (h≠k) group (exemplary embodiments shown in FIG. 1C, with T=7 and 13). A triangular patch is then generated from the constructed triangle. For example, FIG. 1D shows the triangular patch 80 for the Goldberg triangle having T=9 vertices with (h,k)=(3,0).
Each triangular facet of a regular tetrahedron, octahedron, or icosahedron is then decorated with the selected Goldberg triangle. FIG. 1E (left) shows the Goldberg triangle 80 on the faces of a tetrahedron 82, FIG. 1E (center) shows the Goldberg triangle 80 on the faces of an octahedron 84, and FIG. 1E (right) shows the Goldberg triangle 80 on the faces of an icosahedron 86. Finally edges 81 are added that connect vertices across the boundaries of the faces, as illustrated for each of these polyhedral in FIG. 1F.
The resulting tetrahedral cage has 4 T trivalent vertices, sixteen 6 gonal faces, and four triangular faces. The resulting octahedral cage has 8 T trivalent vertices, thirty-two 6 gonal faces, and six square corner faces. The resulting icosahedral cage has 20 T trivalent vertices, eighty hexagonal faces, and twelve pentagonal faces. However, with unequal edge lengths, these cages are not equilateral. With nonplanar faces these cages are not polyhedra and thus not convex.
For T=1 and T=3 we can transform these cages such that all edge lengths are equal and all interior angles in the hexagons are equal. For T=1 this method produces three of the Platonic solids: the tetrahedron, the cube, and the dodecahedron. For T=3, this method produces three of the Archimedean solids: the truncated tetrahedron, the truncated octahedron, and the truncated icosahedron. These cages are geometrically polyhedral because their faces are planar. They are also convex.
Can similar symmetric convex equilateral polyhedra be created from Goldberg triangles for T>3? The present inventors have proven that no such polyhedra are possible if the transformation also requires equiangularity. Even if the transformation does not enforce equiangularity, the resulting “merely equilateral” cages would typically have nonplanar hexagonal faces, and therefore are not polyhedral. Moreover, the nonplanar hexagons defined by the cages are either “boat” shaped or “chair” shaped, and therefore the cages are not convex.
The present inventors found that the difference—convex polyhedral cages with planar hexagons for T=1 and T=3, but non-polyhedral cages with nonplanar faces for T>3—is due to the presence of edges with dihedral angle discrepancy (“DAD”), which is discussed in more detail herein. However, surprisingly the inventors discovered that it is possible to null all of the DADs and thus to create an entirely new class of equilateral convex polyhedra with polyhedral symmetry that we call “Goldberg polyhedra.”
The resulting Goldberg polyhedra and corresponding Goldberg cages may be used, for example, to construct an efficient and nearly spherical framework or dome for enclosing space wherein the edges or struts of the framework are of equal length. Near-spherical convex, equilateral polyhedral structures, and methods for designing such structures, are disclosed that are suitable for enclosing a space, including, for example, a living space, a storage space, a utility space, or the like. The new equilateral cages and/or Goldberg polyhedra may also be used for other purposes such as a providing nearly spherical (e.g., hemispherical, spherical sections, or the like) constructs that may be used as supports. An advantage of such structures is the equilaterality. For example, an equilateral cage will have struts that are all of equal length, so the struts may be fully interchangeable, thereby simplifying manufacture and assembly.